The lifespans of seals in a particular zoo are normally distributed. The average seal lives $15.3$ years; the standard deviation is $3.6$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a seal living between $22.5$ and $26.1$ years.
Solution: $15.3$ $11.7$ $18.9$ $8.1$ $22.5$ $4.5$ $26.1$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $15.3$ years. We know the standard deviation is $3.6$ years, so one standard deviation below the mean is $11.7$ years and one standard deviation above the mean is $18.9$ years. Two standard deviations below the mean is $8.1$ years and two standard deviations above the mean is $22.5$ years. Three standard deviations below the mean is $4.5$ years and three standard deviations above the mean is $26.1$ years. We are interested in the probability of a seal living between $22.5$ and $26.1$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the seals will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the seals will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of seals between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular seal living between $22.5$ and $26.1$ years is $\color{orange}{2.35\%}$.